<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">{Knaster continua}

By a
\e{Knaster continuum}
is meant a
continuum homeomorphic to the inverse limit
$\varprojlim(I_n,f_n)$ of a sequence of unit intervals
$I_n=I$ with open, non-homeomorphic bonding maps $f_n$.
Denote by $\mathcal K$ the class of all Knaster continua.


The most popular examples of Knaster continua is the
\e{buckethandle}
(sometimes called by dynamists
the
\e{horseshoe}
)  $B_1$, where
$f_n(t)=1-|2t-1|$, and the
\e{double buckethandle}
$B_2$  with
$$ f_n(t)=\begin{cases}
3t&amp; \text{if \ $0\le t\le\frac13$},\\
2-3t&amp; \text{if \ $\frac13\le t\le \frac23$},\\
3t-2&amp; \text{if \ $\frac23\le t\le 1$}.
\end{cases}
$$
 Nice geometric models of $B_1$ and $B_2$ can be described
 in the following way (see \cite[pp. 204--205]{Kuratowski1968a}).
 If $C$ denotes the standard Cantor ternary set in $I$, then
 $B_1$ is homeomorphic to the union of all semi-circles
$$\{\,(x,y)\in\mathbb R^2: (x-\frac12)^2+y^2=r^2, y\ge 0\,\},$$
where $r+\frac12\in C$,
and all semi-circles
$$\{\,(x,y): (x-\frac5{2\cdot3^n})^2+y^2=r^2, y\le 0\,\},$$
for $n\in\mathbb N$, where $r+\frac5{2\cdot3^n}\in C\cap [\frac2{3^n},\frac1{3^{n-1}}]$.

See Figure A.

\FIGURE{A}{buckethandle}

%Figure 14.1.6

If $E$ is the quintary Cantor set in $I$ (real numbers in $I$
which can be written in the enumeration system at base 5 without digits 1 and 3), then
$B_2$ is homeomorphic to the union of all semi-circles
$$\{\,(x,y): (x-\frac7{10\cdot5^n})^2+y^2=r^2, y\le0\,\},$$
for $n\in\mathbb N$ and $r+\frac7{10\cdot5^n}\in E\cap
[\frac2{5^{n+1}},\frac1{5^n}]$ and all semi-circles
$$\{\,(x,y):(x-(1-\frac7{10\cdot5^n}))^2+y^2=r^2, y\ge 0\,\},$$
for $n\ge 0$ and $\frac2{5^{n+1}}\le\frac7{10\cdot5^n}-r\le\frac1{5^n}$.

See Figure B--C.

\FIGURE{B}{double buckethandle}

\FIGURE{C}{double buckethandle}

%Figure 13.1.8


\rostere

\item
If  $w_p:I\to I$ is the map such that $w_p(i/p)=0$ for even
$i$, $w_k(i/p)=1$ for odd $i$, where $i$ is an integer
between 0 and $p\in\mathbb N$, and $w_p$ is linear on each
interval $[i/p,(i+1)/p]$,  then $\mathcal K$ is equal to the
family of all continua homeomorphic to inverse limits
$\varprojlim(I_n,f_n)$, where, for each $n$, $I_n=I$ and
$f_n=w_{p_n}$ for some $p_n$ \cite{Rogers1970a}.  So, each
$K\in\mathcal K$ is determined by  a sequence  $\mathbf{p}
=(p_1,p_2,\dots)$ such that $K= \varprojlim(I_n,w_{p_n})$.
We will denote such $K$ by $K(\mathbf{p})$ and call it a
\e{$\mathbf{p}$-adic Knaster continuum}
. Without
loss of generality, one can assume that all $p_n$'s are
prime.

\item
Each $\mathbf{p}$-adic Knaster continuum $K$ is homeomorphic
to the quotient of the \g{$\mathbf{p}$-adic solenoid} $\Sigma(\mathbf{p})$
under the relation
$$a\sim b\Leftrightarrow a=b\quad \text{or}\quad a=b^{-1}, $$
where $a,b\in \Sigma(\mathbf{p})$ and $b^{-1}$ is the group
inverse of $b$. A point of $K$ is an \g{end point}
of $K$ if and only if it is an image under the quotient map
of the neutral element or the element of order 2 (if such
exists) of the corresponding solenoid (see, e.g.,  \cite[p.
43]{Krupski1984b}).


\item
The class $\mathcal K$ of all Knaster continua is equal to
the class of all \g{arc-like} continua  with one or
two end points having the \g{property of
Kelley} and arcs as proper non-degenerate subcontinua and
which themselves are not arcs \cite{Krupski1984b}. In
particular, all of them are
\g{indecomposable}.

\item
There are $2^{\aleph_0}$  mutually non-homeomorphic members
of $\mathcal K$ \cite{Watkins1982a}; in fact, there is a
subfamily of $\mathcal K$ of cardinality $2^{\aleph_0}$
no member of which is an open image of another one \cite{Debski1985a}.

\item
Any open map between two Knaster continua
$\varprojlim(I_n,f_n)$, $\varprojlim(I_n,g_n)$ is
the uniform limit of open maps which are induced by maps
$(I_n,f_n)\to (I_n,g_n)$ between the inverse sequences
\cite[Theorem 4.8, p. 145]{Eberhart+1999a}.


\item
If $K\in \mathcal K$, then every indecomposable continuum
can be mapped onto $K$ \cite[Corollary, p. 455]{Rogers1970a}.

Moreover, if $X$ is an indecomposable (Hausdorff) continuum,
$a,b\in X$ and $a\ne b$ ($a$ and $b$ belong to different
composants\g{composants} of $X$), then there is a continuous
surjection  $f:X\to B_1$ such that $f(a)\ne f(b)$ ($f(a)$
and $f(b)$ belong to different composants of $B_1$, resp.).
It follows that any indecomposable continuum embeds in the
Cartesian product of copies of $B_1$ \cite[Corollaries 1 and
2, p. 305]{Bellamy1973a}.


\item
If $K_1, K_2\in \mathcal K$  then there are $2^{\aleph_0}$
distinct \g{homotopy types} of maps of $K_1$
onto $K_2$ that map an end point of $K_1$ onto an end point
of $K_2$ \cite{Minc1999a}.

\item
If $K\in \mathcal K$ and $f$ is a \g{monotone
map} of $K$, then $f(K)$ is homeomorphic to $K$
\cite{Krupski1984b}.

\item
There is no exactly 2-to-1 map defined on a Knaster continuum \cite{Debski1992a}.
\item
Knaster continua are \gs{absolute retracts}{absolute retract}
for the class of all \g{hereditarily
unicoherent} continua, i.e., if $K\in\mathcal K$, $X$ is a
hereditarily unicoherent continuum  and $K\subset X$, then
$K$ is a retract of $X$. Equivalently, if $A$ is a closed
subset of a  hereditarily unicoherent continuum $X$, then
every map $f$ from $A$ onto a Knaster continuum can be
extended over $X$ (\cite{Mackowiak1984a} for $K=B_1$ and $X$
a Hausdorff hereditarily unicoherent continuum and
\cite{Charatonik+XXXXa} for arbitrary $K\in\mathcal K$).

\item
Any two points of a plane continuum $X$ can be joined by a
\g{hereditarily decomposable}
subcontinuum if and only if $X$ cannot be mapped onto $B_1$
\cite[Theorem 2, p. 133]{Hagopian1974a}.


\item
Every autohomeomorphism of the buckethandle $B_1$ is
\g{isotopic} to some iterate of the shift
homeomorphism $f:B_1\to B_1,
f(x_1,x_2,\dots)=(x_2,x_3,\dots)$ \cite[Theorem 4.4, p.
204]{Aarts+1991a}.

Moreover, any autohomeomorphism $f$ of a Knaster continuum
$\varprojlim(I_n,w_{p_n})$ is isotopic to a standard
homeomorphism $g$ (defined as a map induced by a map
$(I_n,w_{p_n})\to (I_n,w_{p_n})$ of inverse systems); maps
$f$ and $g$ have the same topological entropy and if the
entropy is positive, then they are
\g{semi-conjugate} \cite[Theorem 4, p.
271]{Kwapisz2001a}.

\item
If $e$ is the end point of $B_1$ and   $S, T$ are composants
of $B_1$ which do not contain $e$, then  there exists a
continuous injection $h:B_1\setminus\{e\}\to
B_1\setminus\{e\}$ such that $h(S)=T$ \cite[Theorem
5.1]{Aarts+1991a}.

Any two composants of $B_1$ which do not contain $e$ are homeomorphic \cite{Bandt1994a}.

\item
Every autohomeomorphism of the buckethandle continuum has at
least two \gs{fixed points}{fixed point} \cite{Aarts+1998a}.

\item
If $A$ is a Borel subset of $K\in\mathcal K$ and $A$ is the
union of a family of composants of $K$, then $A$ is
\g{meager} or \g{comeager} \cite{Emeryk1980a},
\cite{Krasinkiewicz1974a}.

\item
Each composant $C$ of $B_1$ different from the
one containing the end point is internal, i.e.,
every continuum $L\subset \mathbb R^2$ intersecting
both $C$ and $\mathbb R^2\setminus B_1$ must intersect
all composants of $B_1$ \cite[Theorem, p. 261]{Krasinkiewicz1974b}.





\item
Any Knaster continuum $K=\varprojlim(I_n,f_n)$, where
$f_m=f_n$ for each $m,n$, is homeomorphic to the
\g{attracting set} of a homeomorphism $h:D^2\to
h(D^2)\subset D^2$ of the plane closed disk
\cite{Barge1986a}. If $f_n(t)=1-|2t-1|$ for all $n$ (i.e.,
$K=B_1$), then $h$ can be defined as a diffeomorphism  which
is called a \h{horseshoe map}---it was first described by S.
Smale in \cite{Smale1965a} (see also \cite{Barge1988a}). See
Figure D.

\FIGURE{D}{the horse-shoe map $f$ maps the stadium shaped region $D$
($A \cup R \cup B$) in such a way, that it shrinks $R$
vertically, stretches $R$ horizontally, contracts the
semicircles then folds the space once and places this result
into itself so that $f(A)$ and $f(B)$ are in the interior of
$A$ and $f(R)$ is in the interior of $D$}

%Include a picture of the horseshoe map as, e.g., in
%J. A. Kennedy, A brief history of indecomposable continua, Continua with the Houston problem book, Dekker, 1995, p. 117, Figure 6.
%or in R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989,
%Figure 3.2, p. 182


\item
The buckethandle $B_1$ and all Knaster continua with two end
points admit no \g{mean} \cite{IllanesXXXXa},
\cite[Theorem 2.2, p. 99]{Kawamura+1996a}.

\item
The \g{hyperspace} $C(K)$ of all subcontinua of
any Knaster continuum $K$ is homeomorphic to the
\g{cone} $Cone(K)$ over $K$ by a homeomorphism
$h:C(K)\to Cone(K)$ which sends $K$ onto the vertex of
$Cone(K)$ and the set of all singletons of $K$ onto the base
of the cone \cite[Corollary 12, p. 639]{Dilks+1981a}.



\rosteree
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